376 CHAPTER24. THEFEYNMANPATHINTEGRAL
andtendtocanceleachotherout.Theexceptionisforpathsinthevicinityofapath
wherethephaseS/ ̄hisstationary. For pathsinthatvicinity,thecontributionsto
thefunctionalintegralhavenearlythesamephase,andhencesumupcontructively.
Butthepathwherethephaseisstationaryisthepathxcl(t′)suchthat
(
δS
δx(t)
)
x=xcl
= 0 (24.51)
Aswehaveseen,thisisjustthecondition thatthepathxcl(t)isasolution ofthe
classical equations of motion. Therefore, for S >> ̄h, the path integralis domi-
natedbypathsintheimmediatevicinityoftheclassicaltrajectoryxcl(t).Inthelimit
̄h→0,onlytheclassicalpath(andpathsinfinitesmallyclosetotheclassicalpath)
contributes. The”semiclassical” orWKBapproximationtotheFeynmanpathin-
tegralis,infact, theapproximationof evaluatingtheintegralintheneighborhood
singleconfigurationxcl(t),i.e.
GT(x,y) =
∫
Dx(t)eiS[x(t)]/ ̄h
≈ prefactor×eiS[xcl(t)]/ ̄h (24.52)
wherethe prefactorisa numerical termwhichcomes fromintegratingoversmall
variationsδxaroundxcl(t).
Problem - Forsome problems, the WKB approximation works extremely well,
evenifS[x,t]isnotsolargecomparedto ̄h. Applythisapproximationtofind the
propagatorofafreeparticle,andcompareittotheexactresult.
24.3 Operators from Path Integrals
Givenatrajectory,x(t),onecanalwaysdefineamomentum,mx ̇(t).Thenanatural
waytodefineamomentumoperatoractingonawavefunctionattimetf isinterms
ofthepath-integral
p ̃ψ(xf,tf)≡
∫
dy
∫
Dx(t)mx ̇(tf)eiS/ ̄hψ(y,t 0 ) (24.53)
wherethepathsrunfrom(x 0 ,t 0 )to(xf,tf),andψ(x,t 0 )isthewavefunctionatany
earliertimet 0 <tf. Letustaket=tfandt 0 =t−!,andthengotothelimit!→0.
Inthatcase
p ̃ψ(x,t) ≡ lim
!→ 0
∫
dym
(x−y)
!
G!(x,y)ψ(y,t−!)
= lim
!→ 0
∫
dym
(x−y)
!
( m
2 πi! ̄h
) 1 / 2
exp
[
i
m
2! ̄h
(x−y)^2 −i
!
h ̄
V(x)
]
ψ(y,t−!)
= lim
!→ 0
∫
dym
(x−y)
!
(
m
2 πi! ̄h
) 1 / 2
exp
[
i
m
2! ̄h
(x−y)^2
]
ψ(y,t−!) (24.54)